Robert Granger

High security pairing-based cryptography revisited

The security and performance of pairing based cryptography has provoked a large volume of research, in part because of the exciting new cryptographic schemes that it underpins. We re-examine how one should implement pairings over ordinary elliptic curves for various practical levels of security. We conclude, contrary to prior work, that the Tate pairing is more efficient than the Weil pairing for all such security levels. This is achieved by using efficient exponentiation techniques in the cyclotomic subgroup backed by efficient squaring routines within the same subgroup.

Hardware and software normal basis arithmetic for pairing-based cryptography in characteristic three

Although identity-based cryptography offers a number of functional advantages over conventional public key methods, the computational costs are significantly greater. The dominant part of this cost is the Tate pairing, which, in characteristic three, is best computed using the algorithm of Duursma and Lee. However, in hardware and constrained environments, this algorithm is unattractive since it requires online computation of cube roots or enough storage space to precompute required results. We examine the use of normal basis arithmetic in characteristic three in an attempt to get the best of both worlds: an efficient method for computing the Tate pairing that requires no precomputation and that may also be implemented in hardware to accelerate devices such as smart-cards.

On Small Characteristic Algebraic Tori in Pairing-Based Cryptography

The value of the Tate pairing on an elliptic curve over a finite field may be viewed as an element of an algebraic torus. Using this simple observation, we transfer techniques recently developed for torus-based cryptography to pairing-based cryptography, resulting in more efficient computations, and lower bandwidth requirements. To illustrate the efficacy of this approach, we apply the method to pairings on supersingular elliptic curves in characteristic three.

Faster squaring in the cyclotomic subgroup of sixth degree extensions

This paper describes an extremely efficient squaring operation in the so-called ‘cyclotomic subgroup’ of

A Comparison of CEILIDH and XTR

\mathbb{F}_{q^6}^{\times}

On the discrete logarithm problem on algebraic tori

, for

On the Function Field Sieve and the Impact of Higher Splitting Probabilities

q \equiv 1 \bmod{6}

On the Static Diffie-Hellman Problem on Elliptic Curves over Extension Fields

. Our result arises from considering the Weil restriction of scalars of this group from

Function Field Sieve in Characteristic Three

\mathbb{F}_{q^6}

Estimates for Discrete Logarithm Computations in Finite Fields of Small Characteristic

to